Unformatted text preview: the endpoints. Since f is bounded, there exists m,M ∈ R so that m ≤ f ( x ) ≤ M for all x ∈ [ a,b ]. Let ε > 0 be given. We set δ = ε/ 2( Mm ). Since f is continuous on [ a,bδ ], there is a partition P of [ a,bδ ] so that U ( P,f )L ( P,f ) < ε/ 2 . As a partition for [ a,b ], we set ˜ P = P ∪ { b } . Then, U ( ˜ P,f )L ( ˜ P,f ) = U ( P,f )L ( P,f ) + ( M *m * ) δ where M * ,m * are the sup and inf of f over [ bδ,b ] respectively. Then, U ( ˜ P,f )L ( ˜ P,f ) < ( ε/ 2) + ( Mm ) δ = ε. Since ε > 0 was arbitrary, this shows that f ∈ R [ a,b ] as desired. 1...
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 Spring '10
 MetCalfe
 Fundamental Theorem Of Calculus, Mean Value Theorem, Value Theorem

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